A high-order discontinuous Galerkin solver for 3D aerodynamic turbulent flows

Transcription

1 A high-order discontinuous Galerkin solver for 3D aerodynamic turbulent flows F. Bassi, A. Crivellini, D. A. Di Pietro, S. Rebay Dipartimento di Ingegneria Industriale, Università di Bergamo CERMICS-ENPC Dipartimento di Ingegneria Meccanica - Università di Brescia Eccomas CFD 2006 Sept. 5 8, Egmond aan Zee, NL

2 Presentation outline Governing equations Compressible RANS + k-ω equations Realizability constraints Numerical solution DG space discretization Time integration Results Stanitz elbow ONERA M6 wing Conclusions

3 Compressible RANS + k-ω equations Compressible RANS + k-ω equations RANS + k-ω equations ( ω = log ω) ρ t + (ρu j ) = 0 x j t (ρu i) + (ρu j u i ) = p + τ ji x j x i x j t (ρe 0) + (ρu j h 0 ) = u i [u i τ ij q j ] τ ij + β ρke eωr x j x j x j t (ρk) + (ρu j k) = [ (µ + σ µ x t ) k ] u i + τ ij j x j t (ρ ω) + (ρu j ω) = x j x j x j [ (µ + σµ t ) ω x j + (µ + σµ t ) ω x k ω x k β ρke eωr x j ] + α k τ u i ij βρe eωr x j

4 Compressible RANS + k-ω equations Compressible RANS + k-ω equations RANS + k-ω equations ( ω = log ω) ρ t + (ρu j ) = 0 x j t (ρu i) + (ρu j u i ) = p + τ ji x j x i x j t (ρe 0) + (ρu j h 0 ) = u i [u i τ ij q j ] τ ij + β ρke eωr x j x j x j t (ρk) + (ρu j k) = [ (µ + σ µ x t ) k ] u i + τ ij j x j t (ρ ω) + (ρu j ω) = x j x j x j [ (µ + σµ t ) ω x j + (µ + σµ t ) ω x k ω x k β ρke eωr x j ] + α k τ u i ij βρe eωr x j

5 Compressible RANS + k-ω equations Compressible RANS + k-ω equations Pressure and heat flux vector p = (γ 1)ρ (e 0 u k u k /2), ( µ q j = Pr + µ ) t h Pr t x j Mean strain-rate, molecular, and turbulent stress tensors S ij = 1 ( ui + u ) j, Z ij = S ij 1 u k δ ij 2 x j x i 3 x k τ ij = 2µ t Z ij 2 3 ρkδ ij, τ ij = 2µZ ij + τ ij Turbulence kinetic energy and viscosity coefficient k = max (0, k), µ t = α ρke eωr

6 Realizability constraints Realizability constraints The variable ω = log ω has been used since the near wall distribution of ω is easier to compute than that of ω The limited value k is introduced to deal with possible negative values of k (the solution of the k equation could take negative values) The variable ω r = max( ω r0, ω) is introduced to preclude too small values of ω ω r0, the minimum allowed value of ω r, is determined by enforcing the following realizability conditions Positive normal turbulent stresses ρu 2 i 0 Schwartz inequality for the shear turbulent stresses 2 ρu i u 2 j ρu i ρu 2 j

7 Realizability constraints Realizability for the k-ω model Realizability constraints in terms of the modeled turbulent stresses ( 2µ t S ij ) ρk 2µ tz ii 0, i = 1, 2, 3 ( 2 ) ( 2 ) 3 ρk 2µ tz ii 3 ρk 2µ tz jj i, j = 1, 2, 3, i j Dividing by µ t and recalling that ρk/µ t = e eωr /α (because of the definition µ t = α ρke eωr ) we obtain ( e eω r ) 2 3 (Z ii + Z e eωr jj) α e eωr α 3Z ii 0, i = 1, 2, 3 α +9 ( Z ii Z jj S 2 ij ) 0, i, j = 1, 2, 3, i j

8 Realizability constraints ω r0 High-Reynolds k-ω model Let s a denote the minimum value of e eωr /α allowed by the preceding inequalities, i.e. the maximum root of ( e eω r ) 2 3 (Z ii + Z e eωr jj) α e eωr α 3Z ii = 0, i = 1, 2, 3 α +9 ( Z ii Z jj S 2 ij ) = 0 i, j = 1, 2, 3, i j Standard k-ωmodel: The minimum value ω r0 is trivially determined since α is a constant parameter, i.e. ω r0 = log (α a)

9 Realizability constraints ω r0 Low-Reynolds k-ω model Low-Reynolds k-ωmodel: in this case α = α t α 0 + Re t/r k 1 + Re t /R k where α t, α 0 and R k are constants parameters but Re t = k/(e eωr ν) The minimum value ω r0 is obtained from the solution of the quadratic equation [ e 2eωr0 αt α0a k ] e eωr0 αt R k ν k R k ν a = 0 where a is the maximum root as computed in the High-Reynolds model case

10 DG space discretization Numerical approximation Differential form u t + F c(u) + F v (u, u) + s(u, u) = 0 Weak form φ u Ω t dx φ F(u, u) dx + φf(u, u) n dσ Ω Ω + φs(u, u) dx = 0 Triangulation T h = {K} of an approximation Ω h of Ω Components of the approximate solution u h on T h u hi Φ h def = { φ h L 2 (Ω h ) : φ h K P k (K) K T h } Ω

11 DG space discretization Notation We introduce the symbols f and F to denote a generic face of the triangulation and the set of all the faces, respectively Let f K + K for K +, K T h. We define the following trace operators {q} def = q+ + q f 2 P [[q]] def = q + n + + q n n K + n + K These definitions can be extended to boundary faces by replacing the value on the exterior element by the boundary data

12 DG space discretization DG space discretization Discrete counterpart of weak form for an element K T h K u h φ h dx h φ h F(u h, h u h ) dx t K + φ h F(u h K, h u h K ) n dσ + φ h s(u h, u h ) dx = 0 K K

13 DG space discretization DG space discretization Discrete counterpart of weak form for an element K T h K u h φ h dx h φ h F(u h, h u h ) dx t K + φ h F(u h K, h u h K ) n dσ + φ h s(u h, u h ) dx = 0 K Flux function of boundary integral not uniquely defined K

14 DG space discretization DG space discretization Discrete counterpart of weak form for an element K T h K u h φ h dx h φ h F(u h, h u h ) dx t K + φ h F(u h K, h u h K ) n dσ + φ h s(u h, u h ) dx = 0 K Flux function of boundary integral not uniquely defined A consistent and stable discretization of viscous fluxes must account for the effect of interface jumps on h u h K

15 DG space discretization DG space discretization Discrete counterpart of weak form for an element K T h K u h φ h dx h φ h F(u h, h u h ) dx t K + φ h F(u h K, h u h K ) n dσ + φ h s(u h, u h ) dx = 0 K Flux function of boundary integral not uniquely defined A consistent and stable discretization of viscous fluxes must account for the effect of interface jumps on h u h Thus we introduce suitable numerical fluxes and modified gradients K

16 DG space discretization DG space discretization Sum above equation over the elements Using the trace operators the DG dicretization reads: find u h1,..., u hm Φ h such that u h φ h dx h φ h F (u h, h u h + r([[u h ]])) dx K T K t h K T K h + [[φ h ]] F ( u ± h, ( hu h + η f r f ([[u h ]])) ±) dσ f F f + φ h s (u h, h u h + r([[u h ]])) dx = 0 K T h K for all φ h Φ h

17 DG space discretization DG space discretization The lifting operator r f, applied to the jumps of u h componentwise, is defined as the solution of the problem φ h r f (v) dx = {φ h } v dσ, φ h [Φ h ] N, v [ L 2 (f ) ] N Ω h f (1) The function r is related to r f by r(v) def = r f (v) (2) f F Inviscid flux: Godunov or van Leer/Hänel flux vector splitting Viscous flux ( F u ± h, ( hu h + η f r f ([[u h ]])) ±) def = {F (u h, h u h + η f r f ([[u h ]]))} (3)

18 Time integration Time Integration The DG space discretized equations can be written as M du dt + R (U) = 0 where U is the global vector of DOFs and M is the global block diagonal mass matrix

19 Time integration Time Integration The DG space discretized equations can be written as M du dt + R (U) = 0 where U is the global vector of DOFs and M is the global block diagonal mass matrix Implicit time discretization

20 Time integration Time Integration The DG space discretized equations can be written as M du dt + R (U) = 0 where U is the global vector of DOFs and M is the global block diagonal mass matrix Implicit time discretization Backward Euler method for steady problems Second-order Runge-Kutta scheme for unsteady problems

21 Time integration Backward Euler scheme By linearizing at time level n we obtain [ M t + R ] (Un ) ( U n+1 U n) = R (U n ) U }{{} A where R(Un ) U is the Jacobian matrix of the DG space discretization and A denotes the linear system matrix

22 Time integration Backward Euler scheme By linearizing at time level n we obtain [ M t + R ] (Un ) ( U n+1 U n) = R (U n ) U }{{} A where R(Un ) U is the Jacobian matrix of the DG space discretization and A denotes the linear system matrix The compact DG space discretization here employed minimizes the number of non-zero blocks for each (block) row of A

23 Time integration Backward Euler scheme By linearizing at time level n we obtain [ M t + R ] (Un ) ( U n+1 U n) = R (U n ) U }{{} A where R(Un ) U is the Jacobian matrix of the DG space discretization and A denotes the linear system matrix The compact DG space discretization here employed minimizes the number of non-zero blocks for each (block) row of A To solve the linear system we can use either direct or iterative solvers

24 Time integration Backward Euler scheme By linearizing at time level n we obtain [ M t + R ] (Un ) ( U n+1 U n) = R (U n ) U }{{} A where R(Un ) U is the Jacobian matrix of the DG space discretization and A denotes the linear system matrix The compact DG space discretization here employed minimizes the number of non-zero blocks for each (block) row of A To solve the linear system we can use either direct or iterative solvers Linear algebra and parallelization are handled through PETSc

25 Time integration Implicit Runge-Kutta scheme Second-order scheme (Iannelli and Baker, 1988) [ ] M t + α R (Un ) K 1 = R (U n ) U [ ] M t + α R (Un ) K 2 = R (U n + βk 1 ) U U n+1 U n = Y 1 K 1 + Y 2 K 2, where the constants α, β, Y 1, Y 2 are given by α = 2 ( ) 2 1, β = 8α 2 2 α, Y 1 = 1 1 8α, Y 2 = 1 Y 1

26 Time integration Implicit Runge-Kutta scheme Second-order scheme (Iannelli and Baker, 1988) [ ] M t + α R (Un ) K 1 = R (U n ) U [ ] M t + α R (Un ) K 2 = R (U n + βk 1 ) U U n+1 U n = Y 1 K 1 + Y 2 K 2, where the constants α, β, Y 1, Y 2 are given by α = 2 ( ) 2 1, β = 8α 2 2 α, Y 1 = 1 1 8α, Y 2 = 1 Y 1 For Y 1 = 1, Y 2 = 0, α = 1 we obtain the backward Euler scheme For Y 1 = 1, Y 2 = 0, α = 1/2 we obtain the Crank-Nicolson scheme

27 Stanitz elbow Stanitz elbow Rectangular cross section duct for secondary flows studies Mis = 0.26 Rew,is = 2,738,800 based on channel width w Tu = 0.01, µt/µ = 0.1 Computational grid hexahedral elements hmin /w = Maximum aspect ratio is 2,332 Computational details Restarted GMRES(30) linear solver with 60 iterations DG-P0: ILU prec., 200 time steps, CFL number up to , residuals to machine zero DG-P1: ILU prec., 200 time steps, CFL number up to , residuals to DG-P2: block diag. prec., 200 time steps, CFL number limited to , residuals to

28 Stanitz elbow Stanitz elbow (a) Symmetry plane Mach contours, DG-P 2 solution. (b) Total pressure loss contours, DG-P 2 solution. Figure: Geometry and surface grid.

29 Stanitz elbow Stanitz elbow Figure: Pressure coefficient on suction and pressure walls for two spanwise sections.

30 Stanitz elbow Stanitz elbow Figure: Total pressure loss contours at exit, exp. and DG-P 1 solution.

31 Stanitz elbow Stanitz elbow Figure: Total pressure loss contours at exit, exp. and DG-P 2 solution.

32 Stanitz elbow Stanitz elbow Figure: Detail of total pressure loss contours at exit, exp. and DG-P 2 solution.

33 ONERA M6 wing ONERA M6 wing Validation case for transonic external flow around a wing M = α = 3.06 Rec, = based on the chord c = m Tu = 0.001, µt/µ = 0.1 Computational grid NPARC alliance grid containing 294,912 hexahedral elements hmin /c = Computational details Restarted GMRES(30) linear solver with 60 iterations for P0 and P 1 solutions, Richardson method with 15 iterations for P 2 solution DG-P0: ILU prec., 300 time steps, CFL number up to , residuals to DG-P1: ILU prec., 400 time steps, CFL number up to , residuals to DG-P2: block diag. prec., 800 time steps, CFL number limited to 15, residuals to

34 ONERA M6 wing ONERA M6 wing (a) Surface grid. (b) Pressure coefficient contours, DG-P 2 solution. Figure: ONERA M6 wing.

35 ONERA M6 wing ONERA M6 wing Figure: Pressure coefficient on the wing surface at different spanwise sections.

36 ONERA M6 wing ONERA M6 wing Figure: Pressure coefficient on the wing surface at different spanwise sections.

37 ONERA M6 wing ONERA M6 wing Figure: Pressure coefficient on the wing surface at different spanwise sections.

38 ONERA M6 wing ONERA M6 wing Figure: Pressure coefficient on the wing surface at different spanwise sections.

39 ONERA M6 wing ONERA M6 wing (a) (b) Spanwise sections. Figure: Turbulence intensity contours, DG-P 2 solution.

40 ONERA M6 wing ONERA M6 wing C L C D C Dp C Df DG-P DG-P DG-P Table: Lift and drag coefficients of ONERA M6 wing.

41 Conclusions The paper presented a parallel, high-order implicit DG solver for the RANS and k-ω equations The code proved reliable, robust and accurate for the complex 3D high-reynolds number flows here considered On-going work Improvement of computational efficiency of the implicit DG method Reliable and accurate shock capturing techniques for high-reynolds number flows Assessment of parallel efficiency of the code... further work promised within the ADIGMA project